$\begingroup$As VictorHenry points out, we need more information to solve this puzzle. Is it guaranteed that at least one of $A,B$ or $C$ committed the crime?$\endgroup$
– Mike EarnestMay 11 '15 at 19:07

$\begingroup$It says, he(inspector) knows exactly that one of three persons is culprit and each person one time says true and one time lies$\endgroup$
– lesaMay 11 '15 at 19:10

2

$\begingroup$The answer to the first question"Can the inspector figure out the culprit?" is "no" unless the inspector also knows that "Each person lies one time and tells the truth the other time." If he does, he should be able to solve the crime before interviewing B and C.$\endgroup$
– Theodore NorvellMay 13 '15 at 13:03

8 Answers
8

If we assume that there are no contradictions in the puzzle (i.e. there cannot be a situation where the "1 lie, 1 truth" rule is violated), then we don't even need to read what B and C say. Only A's statement matters.

If A's statements are Lie, Truth respectively, then his combined statement is that he did it (opposite of first statement) and B did it. This is clearly impossible, so it must be the case that A's statements are Truth, Lie respectively. This means A didn't do it, B didn't do it, so C must have done it. B and C's statements are irrelevant.

$\begingroup$You also has to show that the system is consistent. So both B's and C's statement has to contain one lie and one thruth if C did it. There is a catch in B's statements.$\endgroup$
– TaemyrMay 12 '15 at 7:31

4

$\begingroup$Well, B is lying about knowing C did it. Doesn't mean he's not coincidentally guessing correctly. Similarly, since C is lying about having done it, he telling the truth about B not knowing (for sure) who it was.$\endgroup$
– FlaterMay 12 '15 at 9:03

$\begingroup$You can edit to remove the last line now since the assumption was added to the question criteria.$\endgroup$
– starsplusplusMay 12 '15 at 13:03

If A committed the crime, then both of A's statements are false. This contradicts the rules of the riddle.

If A didn't do it:
I was not is TRUE
B did it should be FALSE.

At this time, we cannot confirm or deny this statement.

If B did it:
I was not is FALSE
I know that C did it. should be TRUE but is FALSE

Much like A, B cannot give two false statements. If B committed the crime, then B would know that B did it, thus making the second statement false.

If B didn't it:
I was not is TRUE
I know that C did it. should be FALSE

Again, we don't entirely know if B's second statement is false or not.

If C did it:
I was not is FALSE
B do not know who was it. is TRUE

Ok, so this is where things get interesting. If C is responsible, that we know that the second statement is true. That also means that the second case for B needs to be true as well. This means that B doesn't know who did it.

NOTE

This does not mean that it isn't C. It only means that B doesn't know C did it. He's just a lucky guesser, that's all.

This also means that A's second case should become true as well. A stated that B did it. We know, in this case, that is false. This means that 'A didn't do it' works out logically.

If C didn't do it:
I was not is TRUE
B do not know who was it. could be TRUE

Obviously C's claim of innocence is true. This means that C's statement that B is wrong must be FALSE. If that is the case, then B must be telling the truth on that line and lying on the first. We have already logically concluded that is contradictory and as such impossible.

Therefore, the only logically valid statement of the six above is C did it

First assume A did it. Then A's statements "I did not" and "B did it" are both false. Contradiction, so A didn't do it.

Now assume B did it. Then A's statements "I did not" and "B did it" are both true. Contradiction, so B didn't do it.

The only possibility remaining is that C did it. Now A's 1st statement and B's 1st statement are true while A's first statement and C's first statement are false, so we can also deduce that B doesn't know who did it.

A says: I did not do it [TRUE]. B did it [FALSE].
B says: I did not do it [TRUE]. I know that C did it [FALSE](just a guess).
C says: I did not do it [FALSE](which means C's the culprit). B does not know who it was [TRUE](which means that B was just guessing).

suppose C's first statement is true, second must be false....... that leads B's first statement is true, second is false....... that leads first statement is true, second is false........ That gives no culprit.

But An inspector knows that exactly one of 3 suspects committed a crime(I won't say, 'none of them')

that clearly means C's first statement is false, and second is true.... that leads B's first statement is true, second is false...... that leads A's first statement is true, second is false. That is it.

Thank you for your interest in this question.
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